The local calculation is a classical result and can be found in [1]}. Consider the topological group \(GL(n, )\) , and let
\(BGL(n)\)  be its classifying space. Denote by \(pEGL(n) \rightarrow BGL(n)\)  the tautological principal \(GL(n)\) -bundle over the classifying
space. We will also write \(\mathop {\mathrm {sk}}_d BGL(n)\)  for the \(d\) -dimensional skeleton of \(BGL(n)\) , i.e. the subspace formed by all cells of
dimension up to \(d\) . Let \(X_d = p^{-1}(\mathop {\mathrm {sk}}_d BGL(n))\)  be the fiber of the tautological bundle over the \(d\) -dimensional skeleton.
